Optimal. Leaf size=60 \[ \frac {\tanh ^{-1}\left (\frac {1-2 b x}{\sqrt {1-4 a b}}\right )}{\sqrt {1-4 a b}}-\frac {\tanh ^{-1}\left (\frac {2 b x+1}{\sqrt {1-4 a b}}\right )}{\sqrt {1-4 a b}} \]
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Rubi [A] time = 0.07, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1161, 618, 206} \[ \frac {\tanh ^{-1}\left (\frac {1-2 b x}{\sqrt {1-4 a b}}\right )}{\sqrt {1-4 a b}}-\frac {\tanh ^{-1}\left (\frac {2 b x+1}{\sqrt {1-4 a b}}\right )}{\sqrt {1-4 a b}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1161
Rubi steps
\begin {align*} \int \frac {a+b x^2}{a^2+(-1+2 a b) x^2+b^2 x^4} \, dx &=\frac {\int \frac {1}{\frac {a}{b}-\frac {x}{b}+x^2} \, dx}{2 b}+\frac {\int \frac {1}{\frac {a}{b}+\frac {x}{b}+x^2} \, dx}{2 b}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1-4 a b}{b^2}-x^2} \, dx,x,-\frac {1}{b}+2 x\right )}{b}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1-4 a b}{b^2}-x^2} \, dx,x,\frac {1}{b}+2 x\right )}{b}\\ &=\frac {\tanh ^{-1}\left (\frac {1-2 b x}{\sqrt {1-4 a b}}\right )}{\sqrt {1-4 a b}}-\frac {\tanh ^{-1}\left (\frac {1+2 b x}{\sqrt {1-4 a b}}\right )}{\sqrt {1-4 a b}}\\ \end {align*}
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Mathematica [B] time = 0.20, size = 138, normalized size = 2.30 \[ \frac {\frac {\left (\sqrt {1-4 a b}+1\right ) \tan ^{-1}\left (\frac {b x}{\sqrt {a b-\frac {1}{2} \sqrt {1-4 a b}-\frac {1}{2}}}\right )}{\sqrt {2 a b-\sqrt {1-4 a b}-1}}+\frac {\left (\sqrt {1-4 a b}-1\right ) \tan ^{-1}\left (\frac {\sqrt {2} b x}{\sqrt {2 a b+\sqrt {1-4 a b}-1}}\right )}{\sqrt {2 a b+\sqrt {1-4 a b}-1}}}{\sqrt {2-8 a b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 164, normalized size = 2.73 \[ \left [-\frac {\sqrt {-4 \, a b + 1} \log \left (\frac {b^{2} x^{4} - {\left (6 \, a b - 1\right )} x^{2} + a^{2} - 2 \, {\left (b x^{3} - a x\right )} \sqrt {-4 \, a b + 1}}{b^{2} x^{4} + {\left (2 \, a b - 1\right )} x^{2} + a^{2}}\right )}{2 \, {\left (4 \, a b - 1\right )}}, \frac {\sqrt {4 \, a b - 1} \arctan \left (\frac {b x}{\sqrt {4 \, a b - 1}}\right ) + \sqrt {4 \, a b - 1} \arctan \left (\frac {{\left (b^{2} x^{3} + {\left (3 \, a b - 1\right )} x\right )} \sqrt {4 \, a b - 1}}{4 \, a^{2} b - a}\right )}{4 \, a b - 1}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 51, normalized size = 0.85 \[ \frac {\arctan \left (\frac {2 \, b x + 1}{\sqrt {4 \, a b - 1}}\right )}{\sqrt {4 \, a b - 1}} + \frac {\arctan \left (\frac {2 \, b x - 1}{\sqrt {4 \, a b - 1}}\right )}{\sqrt {4 \, a b - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.87 \[ \frac {\arctan \left (\frac {2 b x -1}{\sqrt {4 a b -1}}\right )}{\sqrt {4 a b -1}}+\frac {\arctan \left (\frac {2 b x +1}{\sqrt {4 a b -1}}\right )}{\sqrt {4 a b -1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 55, normalized size = 0.92 \[ \frac {\mathrm {atan}\left (\frac {b\,x}{\sqrt {4\,a\,b-1}}\right )+\mathrm {atan}\left (\frac {\frac {3\,x\,\left (4\,a\,b-1\right )}{4}-\frac {x}{4}+b^2\,x^3}{a\,\sqrt {4\,a\,b-1}}\right )}{\sqrt {4\,a\,b-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.46, size = 117, normalized size = 1.95 \[ - \frac {\sqrt {- \frac {1}{4 a b - 1}} \log {\left (- \frac {a}{b} + x^{2} + \frac {x \left (- 4 a b \sqrt {- \frac {1}{4 a b - 1}} + \sqrt {- \frac {1}{4 a b - 1}}\right )}{b} \right )}}{2} + \frac {\sqrt {- \frac {1}{4 a b - 1}} \log {\left (- \frac {a}{b} + x^{2} + \frac {x \left (4 a b \sqrt {- \frac {1}{4 a b - 1}} - \sqrt {- \frac {1}{4 a b - 1}}\right )}{b} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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